A sensitivity estimate for Boolean functions

Abstract

Abstract A Boolean response to a random binary input of length n can be modeled as a {;0, 1}- valued function v defined on a discrete probability space Ω of all subsets of a finite set of size n. An ω ∈ Ω represents the locations of 1's in the input. For a particular j th location, 1 ≤ j ≤ n, we assume that 1 appears with probability ρ j independently of other locations. Then, for ρ = (ρ 1,…, ρ n ), we will investigate P ρ ( v = 1) as a function of ρ . Using the sharp version of the Khinchin inequality, we give an upper estimate for the ℓ 2 norm of the gradient of P ρ ( v = 1) evaluated at ρ = ( 1 2 ,…, 1 2 ) (cf. (5) below). For monotone functions, the estimate applies also to vector of influences of Boolean functions. We also provide a handy expansion of P (·)( v = 1) based on a Fourier expansion of v (cf. (4) below). Numerical analysis of the bounds leads to the conjecture about the sharp bound that depends on cardinality of the underlying set; the sharp version of the Khinchin inequality is also conjectured.

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